Factorise:
$(x^{3} - 2 x^{2} - 5 x + 6)$

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Solution

Let $f (x) = x^{3} - 2 x^{2} - 5 x + 6$
$\Rightarrow f (1) = 1^{3} - 2 1^{2} - 5 \times 1 + 6 = 1 - 2 - 5 + 6 = 0$
Using synthetic division, we have
$x^{2} - x - 6 = 0$ is the other factor
$(x - 1) (x^{2} - x - 6) = 0$
$= (x - 1) (x^{2} - 3 x + 2 x - 6) = 0$
$= (x - 1) (x (x - 3) + 2 (x - 3)) = 0$
$= (x - 1) (x + 2) (x - 3)$ are the factors of $f (x)$

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